Three-orbifolds with positive scalar curvature

TL;DR

This paper classifies complete 3-orbifolds with positive scalar curvature as connected sums of spherical orbifolds, extending previous geometric results using Ricci flow with surgery.

Contribution

It extends the classification of 3-orbifolds with positive scalar curvature by employing Ricci flow techniques, generalizing prior work to include orbifolds with bounded geometry.

Findings

Orbifolds decompose into sums of spherical 3-orbifolds.

Use of Ricci flow with surgery on complete 3-orbifolds.

Extension of previous classification results.

Abstract

We prove the following result: Let $(\mathcal{O},g_0)$ be a complete, connected 3-orbifold with uniformly positive scalar curvature, with bounded geometry, and containing no bad 2-suborbifolds. Then there is a finite collection $\mathcal{F}$ of spherical 3-orbifolds, such that $\mathcal{O}$ is diffeomorphic to a (possibly infinite) orbifold connected sum of copies of members in $\mathcal{F}$. This extends work of Perelman and Bessi$\grave{e}$res-Besson-Maillot. The proof uses Ricci flow with surgery on complete 3-orbifolds, and are along the lines of the author's previous work on 4-orbifolds with positive isotropic curvature.